A Statistical Approach to Quantum Many-Body Eigenstates near Criticality: Multifractality in Hilbert Space

某些孤立的量子多体系统在其自身的统一动力学下从未达到平衡（通过合适的单粒子观测值来定义）。即使在单粒子水平上，由于多粒子相互作用不能为系统本身提供热浴这一事实，系统在任意长时间内保留了其在局部自由度中编码的初始状态的信息。这种行为反映了系统的多体激发态结构，在希尔伯特空间中表现出局域性。例如，这种情况发生在受强无序（多体局部化）影响的相互作用量子系统中，以及某些经典混沌多体系统中。为了更深入地理解一般多体量子系统中的遍历性、混沌和热化，需要详细研究希尔伯特空间中多体波函数的性质和相关的可观测值。特别是，波函数多重分形似乎是多体系统的一个普遍特征，在这一现象学中起着关键作用。多重分形波函数在热力学极限中占据无限的体积，就像扩展状态一样，但同时也是整个空间的一个消失的部分，继承了局域状态的非遍历性。多体波函数的结构，它们的多重分形波动的量化，以及当系统从遍历到局域流动时，这些波动是如何演变的，这些都还没有完全被理解，然而，这至关重要地决定了物理观测的行为。我们建议使用数值超标准广义多重分形分析结合有限尺度来解决希尔伯特空间中多重分形的研究。这种强大的技术扩大了多重分形分析的适用性，以描述扩展，多重分形和局部状态的区域以及它们之间的缩放。通过我们的创新方法，我们的目标是：(a)给出无序相互作用费米子在希尔伯特空间中的多体定域现象学的清晰视角。我们将提供明确的证据，证明多体本征态存在（或不存在）遍历的、真正局部化的和多重分形的中间相。这将导致一个高精度的相互作用费米子的局部和多重分形相图，作为能量密度，相互作用强度和无序的函数。(b)揭示了Fock空间中无无序状态下相互作用玻色子的多重分形。我们的初步研究揭示了这种系统基态的多重分形。我们将研究多重分形作为粒子间相互作用强度的函数的演化，以及与莫特到超流体相变的关系。通过增加局部化学势的倾斜度，我们将研究激发能谱中鲁棒态（参数孤子）的多重分形性质，并将它们与系统基态的多重分形性质联系起来。

Certain isolated quantum many-body systems never reach equilibrium (defined via suitable single-particle observables) under their own unitary dynamics. Even at the single-particle level, the system retains information about its initial state encoded in local degrees of freedom for arbitrarily long times, due to that fact that many-particle interactions fail to provide a thermal bath for the system itself. Such behaviour is a reflection of the structure of the many-body excited eigenstates of the system, which exhibit localisation in Hilbert space. This occurs for example in interacting quantum systems subjected to strong disorder (many-body localisation), as well as in certain classically chaotic many-body systems. A deeper understanding of ergodicity, chaos and thermalisation in generic many-body quantum systems requires a detailed study of the nature of many-body wavefunctions and related observables in Hilbert space. In particular, wavefunction multifractality seems to be a generic feature of many-body systems that plays a key role in this phenomenology. Multifractal wavefunctions occupy an infinite volume in the thermodynamic limit -like extended states do-, but at the same time a vanishing fraction of the whole space -inheriting the non-ergodicity of localised states. The structure of many-body wavefunctions, and the quantification of their multifractal fluctuations and how these evolve as the system flows from ergodic to localised is not fully understood, yet this determines crucially the behaviour of physical observables. We propose to tackle the study of multifractality in Hilbert space using a numerical beyond-standard generalised multifractal analysis in combination with finite-size scaling. This powerful technique enlarges the applicability of multifractal analysis to describe regions of extended, multifractal and localised states and the scaling between these. With our innovative approach we aim to: (a) Give a clear perspective of the many-body localisation phenomenology in Hilbert space for disordered interacting fermions. We will provide unambiguous evidence of the existence (or not) of ergodic, truly localised, and multifractal intermediate phases of the many-body eigenstates. This will lead to a high-precision phase diagram of localisation and multifractality for interacting fermions, as a function of energy density, interaction strength and disorder.(b) Unravel multifractality in Fock space for interacting bosons in the absence of disorder. Our preliminary investigations reveal multifractality in the ground state of such a system. We will study the evolution of multifractality as a function of the inter-particle interaction strength, in relation with the Mott to superfluid phase transition. By adding a tilt to the local chemical potential, we will investigate the multifractal properties of robust (parametric solitonic) states in the excitation energy spectrum, and relate them to those of the system's ground state.